The purpose of this study is to compare the effectiveness and side effects of standard chemotherapy consisting of the two chemotherapy drugs cisplatin and paclitaxel given by vein compared to the same two chemotherapy drugs given partly by vein and partly into the peritoneal cavity. This is an important question because there is some information to suggest that, at least in some patients, when the chemotherapy is given into the peritoneal cavity, it is more effective. Additional purposes of this study are to determine if the genetic mutations BRCA1 and BRCA2 are present in these cancers by analyzing both normal tissue and tumor specimens. These genetic mutations have been found to be present in many families when breast and ovarian cancer are commonly found and may help to predict the overall prognosis or survival in women with ovarian cancer